Kooperativer Bibliotheksverbund

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Format: 1 Online-Ressource (XIV, 243 p. 117 illus., 116 illus. in color)

Edition: 1st ed. 2024

ISBN: 9783031475115

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-031-47510-8

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-031-47512-2

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-031-47513-9

Language: English

DOI: 10.1007/978-3-031-47511-5

URL: Volltext  (URL des Erstveröffentlichers)

URL: lizenzpflichtig

UID:

Format: 1 online resource (245 pages)

Edition: First edition.

ISBN: 3-031-47511-9

Note: Intro -- Preface -- Contents -- Part I Tailoring for Every Body -- 1 Introduction to Part SPIlinkcolor100I -- 1.1 Convex Polyhedra: Background -- 1.2 Digon-Tailoring -- 1.3 Alexandrov's Gluing Theorem -- 1.4 Several Tailoring Examples -- 1.5 Summary of Part-I Results -- 2 Preliminaries -- 2.1 Geodesic Segments -- Quasigeodesics -- 2.2 Gauss-Bonnet Theorem -- 2.3 Cut Locus -- 2.3.1 Cut Locus: Definition and Properties -- 2.3.2 Star-Unfolding and Cut Locus -- 2.3.3 Fundamental Triangles -- 2.3.4 Cut Locus Partition -- 2.4 Cauchy's Arm Lemma -- 2.5 A Rigidity Result -- 2.6 Vertex-Merging -- 3 Domes and Pyramids -- 3.1 Domes -- 3.2 Cube/Tetrahedron Example -- 3.2.1 Slice → G-domes for Cube/Tetrahedron -- 3.3 Proof: G-dome → Pyramids -- 3.3.1 G-domes → Pyramids for Cube/Tetrahedron -- 4 Tailoring via Sculpting -- 4.1 Slice → G-domes -- 4.2 Pyramid → Tailoring -- 4.2.1 Notation -- 4.3 Sufficiently Close Truncations -- 4.3.1 Pyramid Case -- 4.3.2 General Case -- 4.4 Cube/Tetrahedron: Completion -- 4.4.1 Pyramid Removals -- 4.4.2 Pyramid Reductions by Tailoring -- 4.4.3 Seals -- 4.5 Hexagonal Pyramid Example -- 4.6 Tailoring Is Finer than Sculpting -- 5 Pyramid Seal Graph -- 5.1 Pyramid Digon Removal -- 5.1.1 Notation I -- 5.2 Cone Viewpoint -- 5.2.1 Notation II -- 5.3 Examples -- 5.4 Preliminary Lemmas -- 5.5 Pyramid Seal Graph Is a Tree -- 5.5.1 Other Digon Orderings -- 6 Algorithm for Tailoring via Sculpting -- 6.1 Algorithm 1: Slice → g-domes -- 6.1.1 Complexity of Sculpting -- 6.2 Algorithm 2: g-dome → Pyramids -- 6.3 Algorithm 3: Pyramid → Digons -- 6.4 Overall Tailoring Algorithm -- 7 Crests -- 7.1 Two Examples -- 7.2 Main Theorem and Supporting Lemmas -- 7.3 Algorithm 4: Pyramid → Crest -- 8 Tailoring via Flattening -- 8.1 Proof -- 8.1.1 Digon-Tailor P →Pflat -- 8.1.2 Vertex-Merge Q →Qflat -- 8.1.3 Scale Qflat →Qsflat -- 8.1.4 Trim Pflat →Psflat. , 8.1.5 Reverse Psflat →Qt -- 8.1.6 Theorem: Tailoring via Flattening -- 8.2 Algorithm for Tailoring via Flattening -- 9 Applications of Tailoring -- 9.1 Enlarging and Reshaping -- 9.2 P-unfoldings -- 9.2.1 P-unfoldings and Reshaping -- 9.2.2 P-unfoldings and the WBG Theorem -- 9.3 Continuously Folding P onto Q -- Part II Vertex-Merging and Convexity -- 10 Introduction to Part SPIlinkcolor100II -- 10.1 Unfolding Convex Polyhedra -- 10.2 Part SPIlinkcolor100II Topics and Results -- 11 Vertex-Merging Reductions and Slit Graphs -- 11.1 Slit Graphs for Vertex-Mergings -- 11.2 Example: Reductions of Flat Hexagon -- 11.3 Example: Reductions of Cube -- 11.4 Example: Icosahedron -- 11.5 Example: Hexagonal Shape with Cycle -- 11.6 Vertex-Merging and Unfoldings -- 11.7 Unfolding Irreducible Surfaces -- 11.7.1 S: Doubly Covered Triangle -- 11.7.2 Net and Overlap -- 11.7.3 S: Isosceles Tetrahedron -- 12 Planar Spiral Slit Tree -- 12.1 Sequential Spiral Merge -- 12.2 Notation -- 12.3 Algorithm Description -- 12.4 Planar Proof -- 13 Convexity on Convex Polyhedra -- 13.1 Convex Curves -- 13.2 Notions of Convexity -- 13.3 Ag-Convexity -- 13.4 Geodesic Segments and Convex Sets -- 13.5 Relative Convexity -- 13.6 Convex Hull -- 13.7 Relative Convex Hull -- 13.8 Extreme Points -- 13.9 Relative Convex Hull of Vertices -- 13.10 Summary of Properties -- 13.10.1 Ag-Convexity -- 13.10.2 Geodesic Segments and Convex Sets -- 13.10.3 Relative Convexity -- 13.10.4 Convex-Hull Properties: conv(S) -- 13.10.5 Relative Convex Hull: rconv(S) -- 13.10.6 Extreme Points: ext(S) -- 13.10.7 Relative Convex Hull of Vertices -- 14 Minimal-Length Enclosing Polygon -- 14.1 Properties of the Minimal Enclosing Polygon -- 14.2 Shortening Algorithm -- 14.2.1 Curve-Shortening Flow -- 14.2.2 Algorithm Overview -- 14.2.3 Finding an Enclosing Geodesic Polygon -- 14.2.4 Algorithm for Curve-Shortening. , 14.3 AG-Convexity and Z -- 14.4 Algorithm for rconv(V)= R(W) -- 15 Spiral Tree on Polyhedron -- 15.1 Notation -- 15.2 Icosahedron Example -- 15.3 Spiraling Algorithm for rconv -- 15.4 Proof: Slit Graph is a Tree -- 15.5 Spiraling Algorithm for Z(V)=min[V] -- 16 Unfoldings via Slit Trees -- 16.1 Notation -- 16.2 Unfoldings via Spiraling Algorithms -- 16.2.1 Two Cones -- 16.2.2 Reduction to Cylinder -- 16.2.3 Cube Example -- 16.2.4 Icosahedron Example -- 17 Vertices on Quasigeodesics -- 17.1 Notation -- 17.2 Quasigeodesics Through k Vertices -- 17.2.1 |V(Q)|=1 -- 17.2.2 |V(Q)|=2 -- 17.2.3 |V(Q)|=k, with 3k n -- 18 Conclusions and Open Problems -- 18.1 Part I -- 18.2 Part II -- Symbols -- References -- Index.

Additional Edition: ISBN 3-031-47510-0

Language: English

UID:

Format: XIV, 243 p. 117 illus., 116 illus. in color. , online resource.

Edition: 1st ed. 2024.

ISBN: 9783031475115

Content: The focus of this monograph is converting-reshaping-one 3D convex polyhedron to another via an operation the authors call "tailoring." A convex polyhedron is a gem-like shape composed of flat facets, the focus of study since Plato and Euclid. The tailoring operation snips off a corner (a "vertex") of a polyhedron and sutures closed the hole. This is akin to Johannes Kepler's "vertex truncation," but differs in that the hole left by a truncated vertex is filled with new surface, whereas tailoring zips the hole closed. A powerful "gluing" theorem of A.D. Alexandrov from 1950 guarantees that, after closing the hole, the result is a new convex polyhedron. Given two convex polyhedra P, and Q inside P, repeated tailoring allows P to be reshaped to Q. Rescaling any Q to fit inside P, the result is universal: any P can be reshaped to any Q. This is one of the main theorems in Part I, with unexpected theoretical consequences. Part II carries out a systematic study of "vertex-merging," a technique that can be viewed as a type of inverse operation to tailoring. Here the start is P which is gradually enlarged as much as possible, by inserting new surface along slits. In a sense, repeated vertex-merging reshapes P to be closer to planarity. One endpoint of such a process leads to P being cut up and "pasted" inside a cylinder. Then rolling the cylinder on a plane achieves an unfolding of P. The underlying subtext is a question posed by Geoffrey Shephard in 1975 and already implied by drawings by Albrecht Dürer in the 15th century: whether every convex polyhedron can be unfolded to a planar "net." Toward this end, the authors initiate an exploration of convexity on convex polyhedra, a topic rarely studied in the literature but with considerable promise for future development. This monograph uncovers new research directions and reveals connections among several, apparently distant, topics in geometry: Alexandrov's Gluing Theorem, shortest paths and cut loci, Cauchy's Arm Lemma, domes, quasigeodesics, convexity, and algorithms throughout. The interplay between these topics and the way the main ideas develop throughout the book could make the "journey" worthwhile for students and researchers in geometry, even if not directly interested in specific topics. Parts of the material will be of interest and accessible even to undergraduates. Although the proof difficulty varies from simple to quite intricate, with some proofs spanning several chapters, many examples and 125 figures help ease the exposition and illustrate the concepts.

Note: I. Tailoring for Every Body -- 1. Introduction to Part I -- 2. Preliminaries -- 3. Domes and Pyramids -- 4. Tailoring via Sculpting -- 5. Pyramid Seal Graph -- 6. Algorithms for Tailoring via Sculpting -- 7. Crests -- 8. Tailoring via Flattening -- 9. Applications of Tailoring -- II. Vertex-Merging and Convexity -- 10. Introduction to Part II -- 11. Vertex-Merging Reductions and Slit Graphs -- 12. Planar Spiral Slit Tree -- 13. Convexity on Convex Polyhedra -- 14. Minimal-length Enclosing Polygon -- 15. Spiral Tree on Polyhedron -- 16. Unfolding via Slit Trees -- 17. Vertices on Quasigeodesics -- 18. Conclusions -- Bibliography -- References -- Index.

In: Springer Nature eBook

Additional Edition: Printed edition: ISBN 9783031475108

Additional Edition: Printed edition: ISBN 9783031475122

Additional Edition: Printed edition: ISBN 9783031475139

Language: English

DOI: 10.1007/978-3-031-47511-5

URL: https://doi.org/10.1007/978-3-031-47511-5